Mathematical Modelling of Pattern Formation in Yeast Biofilms

Abstract

We use mathematical modelling and experiments to investigate yeast biofilm growth and pattern formation. Biofilms are sticky communities of cells and fluid residing on surfaces. As yeast biofilms are a leading cause of hospital-acquired infections, researchers have developed methods of growing them on semi-solid agar. These biofilms initially form a thin circular shape, before transitioning to a non-uniform floral morphology. To quantify biofilm growth, we use a radial statistic the measure expansion speed, and an angular pair correlation function to quantify petal formation. These spatial statistics enable comparison between experiments and mathematical model predictions. Our motivation is to improve understanding of the physical mechanisms governing biofilm formation. One hypothesised mechanism is nutrient-limited growth, in which movement and consumption of nutrients drives growth and generates patterns. Another hypothesis is that yeast biofilms expand by sliding motility, where cell proliferation and weak biofilm–substratum adhesion drive growth. Mathematical modelling enables us to investigate the contribution of each hypothesised mechanism to biofilm growth and pattern formation. We use a reaction–diffusion system with non-linear, degenerate cell diffusion to model nutrient-limited biofilm growth. This model admits sharp-fronted travelling wave solutions that advance with constant speed, an assumption consistent with experimental data. To investigate whether the reaction–diffusion model can explain petal formation, we consider the linear stability of planar travelling wave solutions to transverse perturbations. There is good agreement between the theory and experimental data, suggesting that nutrient-limited growth can explain floral pattern formation. Next, we introduce biofilm mechanics by deriving a two-phase fluid model. We treat the biofilm as a mixture of cells and an extracellular matrix, and obtain governing equations from mass and momentum conservation. Since yeast biofilm height is small compared to their radius, we use the thin-film approximation in two scaling regimes to simplify the model. The extensional flow regime involves weak biofilm–substratum adhesion, and models expansion by sliding motility. In contrast, the lubrication regime involves strong biofilm– substratum adhesion, and large pressure and surface tension. We compute axisymmetric numerical solutions to both thin-film models to investigate how mechanics affects biofilm growth. There is good agreement between the extensional flow model and experimental data, suggesting that sliding motility can explain expansion speed. Parameter sensitivity analyses show that increased nutrient supply and biomass production rates generate faster expansion. The effect of surface tension, which represents the strength of cell–cell adhesion, is the key difference between the two regimes. In the extensional flow model, surface tension inhibits ridge formation close to the leading edge, but does not affect expansion speed. In contrast, surface tension generates radial expansion in the lubrication regime. Since the thin-film models enable us to predict biofilm height and nutrient uptake explicitly, they provide a more detailed description of biofilm growth than the reaction–diffusion model. However, their complexity makes it more difficult to use linear stability analysis to investigate two-dimensional patterns. This problem, and alternative expansion mechanisms such as osmotic swelling and agar deformation, provide avenues for future work.